Download Basic Calculus (Limits and Continuity) by: Mrs. AGJRitual

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BASIC
CALCULUS
Ms. Angel Grace R. Jaen
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Let us pray for the strength and
protection of all our frontliners
especially our medical frontliners.
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CHAPTER I
WEEK 1
Limits and Continuity
Evaluate the limits
of the functions by
using tabular and
graphical methods.
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Let’s have an
activity!
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y = 2x+3
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What is the value of y, if x = 2?
A. 7
B. 4
C. 2
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y = 2x+3
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What is the value of y, if x = -2?
A. - 7
B. - 3
C. - 1
7
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𝒙−𝟏
𝒚=
𝒙+𝟑
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What is the value of y, if x = -3?
A. 0
B. undefined
C. 6
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𝒙−𝟏
𝒚=
𝒙+𝟑
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What is the value of y, if x = 1?
A. 0
B. undefined
C. 6
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𝒙+𝟔
𝒚=
𝒙+𝟐
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What is the value of y, if x = 2?
A. 2
B.
1
2
C. 3
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ThetoTabular
Method
Consider the linear function:
f(x) = 2x – 3
Let us say that we want to determine
the limit of f(x) as the values of x
approach to 1. We are interested in
looking at the behavior of the f as the
values of x get closer and closer to the
number 1.
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Table 1 shows the value of x
approaching the number of 1
from the left; that is, the value
of x are getting closer and
closer to 1, but they are all
less than 1. How do the
values of y = f(x) behave as
the result? They also get
closer to the number -1.
x
0
0.5
0.75
0.8
0.9
0.99
0.999
0.9999
F(x) = 2x - 3
-3
-2
-1.5
-1.4
-1.2
-1.02
-1.002
-1.0002
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In Table 2, we make the
values of x approach 1 from
the right. This time, the values
of x are getting closer and
closer to 1, but they are all
greater than 1. Then, we see
that the resulting values of
f(x) also approach the
number -1.
x
2
1.5
1.25
1.2
1.1
1.01
1.001
1.0001
F(x) = 2x - 3
1
0
-0.5
-0.6
-0.8
-0.98
-0.998
-0.9998
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have
edityou
Master
observed?
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x
0
0.5
0.75
0.8
0.9
0.99
0.999
0.9999
F(x) = 2x - 3
-3
-2
-1.5
-1.4
-1.2
-1.02
-1.002
-1.0002
x
2
1.5
1.25
1.2
1.1
1.01
1.001
1.0001
F(x) = 2x - 3
1
0
-0.5
-0.6
-0.8
-0.98
-0.998
-0.9998
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Limits
• Limits are a tool for reasoning about function
behavior, and tables are a tool for reasoning
about limits. One nice thing about tables is
that we can get more precise estimates of
limits than we'd get by eyeballing graphs.
• When using a table to approximate limits, it's
important to create it in a way that simulates
the feeling of getting "infinitely close" to some
desired x-value.
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Limits
• Imagine we're asked to approximate this limit:
𝑥−2
lim 2
𝑥→2 𝑥 − 4
• Note: The function is actually undefined
at x=2, because the denominator evaluates to zero, but
the limit as x approaches 2 still exists.
• Step 1: We'd like to pick a value that's a little bit less
than x=2 (that is, a value that's "to the left" of 2 when
thinking about the standard x-axis), so maybe start with
something like x=1.9.
x
f(x)
1.9
0.2564
2
undefined
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• Step 2: Try a couple more x-values
to simulate the feeling of getting
infinitely close to x=2 from the left.
x
f(x)
1.9
0.2564
1.99
0.2506
1.999
0.25001
2
undefined
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• Step
3:edit
Approach
x=2 style
from the
right just like we did from the left.
We want to do this in a way that
simulates the feeling of getting
infinitely close to x=2.
x
f(x)
•
1.9
0.2564
1.99
1.999
2.0001
0.2506 0.25001 0.24999
Note: We've removed x=2x=2x, equals,
2 from the table to save space, and also
because it isn't necessary for reasoning
about the limit value.
2.01
0.2494
2.1
0.2439
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Let’s have an
activity!
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x
1
1.25
1.5
1.75
1.9
1.99
1.999
1.9999
f(x) = 4 + 2x
6
6.5
7
7.5
7.8
7.98
7.998
7.9998
x
3
2.75
2.5
2.1
2.01
2.001
2.0001
2.00001
lim 𝟒 + 𝟐𝒙 = 𝟖
𝒙→𝟐
f(x) = 4 + 2x
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9.5
9
8.2
8.02
8.002
8.0002
8.00002
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𝒙𝟐 − 𝟏
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𝒇(𝒙) =
0
0.5
0.75
0.8
0.9
0.99
0.999
0.9999
x
𝒙−𝟐
0.5
0.5
0.35
0.3
0.17
0.0197
0.001997
0.00019997
lim
𝒙𝟐 − 𝟏
𝒙→𝟏 𝒙 − 𝟐
1.8
1.75
1.5
1.1
1.01
1.001
1.0001
1.00001
=𝟎
𝒙𝟐 − 𝟏
𝒇(𝒙) =
𝒙−𝟐
-11.2
-8.25
-2.5
-0.23
-0.02
-0.002
-0.0002
-0.00002
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Any question?
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“
Life has no limitations, except
the ones you make.”
- Les Brown
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Thank You!
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